Global existence combined with general decay of solutions for coupled Kirchhoff system with a distributed delay term
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Global existence combined with general decay of solutions for coupled Kirchhoff system with a distributed delay term Nadjat Doudi1 · Salah Boulaaras2,3 Received: 8 July 2020 / Accepted: 23 September 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract The paper deals with the proof of the global existence of solutions for system of nonlinear viscoelastic Kirchhoff system with a distributed delay and general coupling terms in a bounded domain. The current study was performed by using the energy method along with Faedo–Galerkin method and under some suitable conditions in coupling terms parameters. In addition, we prove the stability result by using the multiplier method. Keywords Global existence · Lyapunov functional · Faedo–Galerkin method · General Decay · Kirchhoff system · Distributed delay term · Viscoelastic term Mathematics Subject Classification 35L90 · 74G25 · 35B40 · 26A51
1 Introduction Let H = × (τ1 , τ2 ) × (0, ∞), in the present paper, we are interested in the following nonlinear viscoelastic Kirchhoff system with a distributed delay and general coupling terms
On the occasion of the 80th birthday of the second author’s mother, Mrs. Fatma Bint Al-Tayeb Zeghdoud.
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Salah Boulaaras [email protected] ; [email protected] Nadjat Doudi [email protected]
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Laboratory of Applied Mathematics, ‘LMA’ Mohamed Khider University, Biskra, Box. 145 rp, 07000 Biskra, Algeria
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Department of Mathematics, College of Sciences and Arts, Al-Ras, Qassim University, Buraydah, Kingdom of Saudi Arabia
3
Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Oran, Algeria 0123456789().: V,-vol
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N. Doudi, S. Boulaaras
⎧ t ⎪ l u − M(∇u2 )u − u + ⎪ |u | g1 (t − s)u(s)ds ⎪ t tt tt ⎪ ⎪ 0 ⎪ τ2 ⎪ ⎪ ⎪ ⎪ −k1 u t − μ1 ()u t (x, t − )d + αv = 0, ⎨ τ1 t ⎪ l v − M(∇v2 )v − v + ⎪ |v | g2 (t − s)v(s)ds ⎪ t tt tt ⎪ ⎪ 0 ⎪ τ2 ⎪ ⎪ ⎪ ⎪ μ2 ()vt (x, t − )d + αu = 0, ⎩ −k2 vt −
(1.1)
τ1
where (x, , t) ∈ H. With the initial data and boundary conditions ⎧ (u(x, 0), v(x, 0)) = (u 0 (x), v0 (x)), in ⎪ ⎪ ⎨ (u t (x, 0), vt (x, 0)) = (u 1 (x), v1 (x)), in (u ⎪ t (x, −t), vt (x, −t)) = ( f 0 (x, t), g0 (x, t)), in × (0, τ2 ) ⎪ ⎩ u(x, t) = v(x, t) = 0, in ∂ × (0, ∞)
(1.2)
where be a bounded domain in Rn with smooth boundary ∂, l, α is positive constants. Here, denotes the Laplacien operator, the second integral represents the distributed delay and μ1 , μ2 : [τ1 , τ2 ] → R are a bounded functions, where τ1 , τ2 are two real numbers satisfying 0 ≤ τ1 < τ2 , and g1 , g2 are the relaxation functions. M is a smooth function defined by M : R+ → R+ r → M(r ) = a + br
(1.3) γ
(1.4)
with a, b > 0. In 1876 Kirchhoff proposed an equation named after him, which is a generalization of the D’Alembert equation, as it belongs to the wave equation models, which describe the transverse vibration of a chain fixed at its end. The problem we are studying is a description of the axially moving viscoelastic that consist of t
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